12. Introduction to Relativity

Professor Ramamurti Shankar: So,
let's begin now. First of all,
I'm assuming all of you have some idea what special
relativity means. There are two theories of
relativity, one is the special theory and one is the general
theory. The general theory is something
that we won't do in any detail. Special theory is something we
will do in reasonable detail. So, it's good to begin by
asking some of you what is your present understanding of what
the subject is all about. Yes, sir?
The Yale cap, what do you think it's about?
Student: It's about relative speed in two reference
systems. Professor Ramamurti
Shankar: Okay, it's about relative speed in
two reference systems. I'll come to you;
then I'll come to you. Student: It's based on
the postulate that the laws of physics are the same in any two
references moving in uniform motion relative to one another
and the speed of light is constant in all references.
Professor Ramamurti Shankar: Okay,
I will take the last row there. Student: [inaudible]
Professor Ramamurti Shankar: Okay,
so what I've heard so far is that the laws of physics are the
same for two people who are both in inertial frames of reference
and the velocity of light's a constant.
Right. That's certainly the way we
understand the special relativity theory.
But it's a very old one. It's been going on long before
Einstein came. There was a theory of
relativity at the time of Newton and that's where I want to
begin. Relativity is not a new idea at
all, it's an old one. And the old idea can be
illustrated in this way and it will agree with your own
experience. So, the standard technique for
all of relativity is to get these high speed trains.
I'm going to have our own high speed train;
this is the top view of the train.
And like in everything I do, we'll get away with the lowest
number of dimensions, which happens to be this one
spatial dimension and of course there is time.
So, the train is moving along the x axis.
You are in this train. You board the train and all the
blinds are closed because you don't want to look outside.
That's not because you're traveling through some parts of
New Jersey; you don't want to look outside
for this particular experiment. You get into the train,
you settle down and you explore the world around you.
You pour yourself a drink, you play pool,
you juggle some ping pong balls, tennis balls,
and you have a certain awareness of what's happening,
namely, your understanding of the mechanical world,
and then you go to sleep. When you are sleeping,
some unseen hand gives to this train a large velocity,
200 miles an hour. The question is,
"When you wake up, can you tell if you're moving
or not?" That's the whole question.
Will this speed, whatever I gave you,
200 miles per hour--will it do anything to you in this train
that will betray that velocity? So, when you wake up will you
say I'm moving or not? Now, you might say,
I'm not moving because I'm on Amtrak and I know this train is
not going anywhere. That kind of sociological
reason, by the way, there are many of them,
you cannot invoke. You can only say,
"I'm on this train. Is anything different?"
And the claim is that nothing will be different.
You just will not know you are moving.
Now, if the train picks up speed, or slows down,
you will know right away. If it picks up speed or
accelerates, you find yourself pushed against the back of the
seat or if the driver slams on the brake,
you will slam into the front of the seat in front of you.
No one is saying that when the motion is accelerated,
you will not know. Accelerated motion can be
detected in a closed train without looking outside.
The question is uniform velocity, no matter how high,
can that be perceived? Can that be detected?
So, at the time of Galileo and Newton everybody agreed that you
cannot detect it. Remember that if you started
out and Newton's laws worked for you, you are called an inertial
observer. One of the laws you want is,
if you leave something, it should stay where it is.
When the train is accelerating, that won't be true.
You leave things on the floor when it's accelerating,
things will slide backwards. So, with no apparent force
acting on it, things will begin to
accelerate; that's a non-inertial frame.
We are not interested in that. You started out as an observer
for whom the laws of Newton work, the laws of inertia work,
F = ma, then you go to sleep and you wake up.
So, when I said everything looks the same,
I really meant that the laws of Newton continue to be the same
because if the laws of Newton are the same,
everything will look the same. That's what it means to say
"everything looks the same." Our expectations of what
happens when I throw it up or what happens when two billiard
balls collide, everything is connected to the
laws of Newton. So, the claim is,
the laws of Newton will be unchanged when this velocity is
added on to you. Now, we should be clear about
one thing. If there is a train next to you
in the beginning -- let me just put it on this side for
convenience -- and you got in and you boarded this train but
you looked at this train and it was not moving.
If you lift the blind and look through, you'll see the other
train and there's another passenger in the other train and
you look at each other, you're not moving.
When you wake up after this brief nap, you find when you
look outside the other train is moving at 200 miles an hour.
The question is, "Can you tell if it's you who's
responsible for this relative motion,
or maybe nothing happened to you and the other train is
moving the opposite way?" And the claim of relativity is
that you really cannot tell. You can tell there is motion
between the two trains that wasn't there before.
That's very clear if you look outside but there is no way to
tell what actually happened when you were sleeping.
Whether you were given the velocity of 200 to the right or
the other train was given a velocity of 200 to the left or
maybe a combination of the two, you just cannot tell.
That's the word "relative." So far -- I didn't tell you --
if you have only one train, what I told you earlier,
is that uniform velocity does not leave its imprint on
anything you can measure. If you look outside,
of course you can see the motion of the other train,
but you still cannot tell who is moving.
You cannot distinguish between different possibilities.
So, you have every right to insist that you are not moving
and the other train is moving the opposite way.
Once again, you can make this argument only for uniform
relative motion. If your train is accelerating,
now I'm saying it as if it is an absolute thing,
and it is. You cannot say,
"I'm not accelerating, the other train is accelerating
in the other direction." You cannot say that because
you're the one who is barfing up and throwing up and slamming
your head on the wall; nothing is happening to the
other person. You cannot say "I'm still in
the same frame, you are going the opposite
way." If you are going the opposite
way, why am I throwing up? Or if you are in a rocket and
the rocket's taking off and the G forces are enormous,
many times your weight, it is the astronauts who are
going through the discomfort. At that time they cannot say we
are at rest and everyone is going the opposite way because
no one else is in danger, but they are.
So, accelerated motion will produce effects.
You cannot talk your way out of that.
But uniform velocity will produce no effects on you and no
effects on the other person. You can detect relative motion
but you cannot in any sense maintain that you are moving and
he's not or that he's moving and you are not.
You can say, "I am at rest,
things are the same as before, the train is moving the
opposite way." Now, if you go in the Amtrak
and you look outside and you don't see another train,
but you see the landscape, you see trees and cows and
everything, going at 200 miles an hour in the opposite
direction, you have some reason to believe
that probably the ground is not moving and you are moving.
But that's just based on what I called earlier some sociological
factors. In other words,
it's completely possible to devise an experiment in which
somebody puts the whole landscape on wheels and when you
go to sleep the landscapes, cows and trees are made to move
the opposite way. Not very likely,
but that's because we know in practice no one is going to
bother to do that just to fool you.
But if that did happen, you won't know the difference.
So, the reason we rule that out is we know some extraneous
things not connected to the laws of physics.
That's why we don't like to open the window and look at the
landscape because then we have a bias.
Open the window and look at another train and you just won't
know. That is the principle of
relativity, that uniform motion between two observers,
both of whom are inertial, is relative.
Each one can insist that he or she is not moving;
the other person is the one who is moving.
Of course, now, if the two, in reality,
if the two trains were at rest--Let's imagine my train got
accelerated. So, during the time it was
accelerated, I would know, but if I was sleeping at the
time, I don't know and when I wake up
and the acceleration is gone and the velocity is constant,
that's when I say, "I just cannot tell."
All right. Now, let's show once and for
all that the laws of Newton are not going to be modified.
So, you find the laws of Newton before you go to sleep,
you wake up, you find them again,
you'll get the same laws; that's the claim.
I hope you understand that all the mechanical things you see in
the world around you come just from F = ma.
We have seen projectiles and collision of billiard balls and
rockets; they're all Newtonian mechanics.
So, to say that things will look the same is to say the laws
of Newton that you will deduce before and after waking up will
be the same. So, let's show that.
When you show that, you're really done with it once
and for all. So, let's do the following.
Here is the x axis and here is my frame of reference.
This is my x axis. Let's call this the origin.
The frame goes to negative and positive x values.
Pick some object sitting at the point x.

Now, we are going to first define the notion of an event.
An event is something that happens at a certain place at a
certain time; that's called an event.
For example, if there's a little firecracker
going off somewhere at some time, the x is where it
happened and the t is when it happened.
So, this is space-time. Once again, space-time does not
require Einstein coming in at all.
We have known for thousands of years that if you want to set up
a meeting with somebody, you've got to say where and
you've got to say when and things do happen in space-time.
The fact that you need x and t,
or, if you're living in three spatial dimensions,
the fact that you need x, y,
z and t is not new.
That is not the revolution Einstein created.
The fact that you need four coordinates to label an event is
nothing new. What he did that is new will be
clear later. So, does everyone understand
what an event means? Okay?
An event is something that happens and to say exactly where
and when it happened, in our world of one dimension,
we give it an x and we give it a t.
Now, that's me, and I'm going to give my frame
of reference the name S. It turns out S is not
just based on just my name. This is the canonical name for
two observers, one is called S and one
is called S prime. So, S prime,
let's say, is you. So, your frame of reference is
going to be taken to be sliding relative to mine.
So, let's draw a y axis here.
We don't really deal with the y coordinate but just to
give you a feeling, this is my y axis;
that's my x axis; this is my origin.
y is not going to play a big role.
Now, you are sliding this way, to the right,
and your speed or velocity is always denoted by u.

Some number of meters per second, you are zooming to the
right at some speed u. So, imagine now you are going
past me. At some instant--I'm sitting
here at the origin of coordinates, you cross me and
then a little later you are somewhere there.
So, that's your y frame axis;
that's your origin. And the same event you say has
a coordinate x prime.

We arrange it so that when you zoom past me,
you set your clock to zero and I set my clock to zero.
When you want to set the clock to zero is completely arbitrary.
So, we will decide right when you pass me.
I'm at the origin of my coordinates, you're at the
origin of your coordinates. When you pass me,
we'll click our stop watches and we will set the time to
zero. So, here's an event.
You and I crossed. What are the coordinates for
that event? According to me,
that event occurred at x = 0 and the time was chosen to
be zero. According to you,
your origin was also on top of my origin, so x prime was
zero and the time is just the time.
Everybody has a single time, and that time is called zero.
That is one event. We made the coordinate of the
event zero zero for both you and me.
It is zero in space because my origin crossed your origin.
That crossing took place at my origin;
that's why my x is zero and took place at your origin,
that's where your x prime is zero and the common
time we chose to be zero by convention.
Then, we want a second event. So, let's say the second event
is some firecracker going off here.
Here is something I should explain that I used to forget in
the previous years. When I say I am moving,
I imagine I am part of a huge team of people who are all
moving with me. So, I've got agents all over
the x axis who are my eyes and ears;
they are looking out for me. So, even though I am here,
if there's a firecracker exploding here,
my guys will tell me. And you are carrying your own
agents. Let's say at every point
x you have a reporter, x = 1,2,
3,4; there are people sitting and
watching. So, when I say I see something,
I really mean me and my buddies, all traveling the same
train at the same speed, all over space taking notes on
what's happening. We'll pool our information
later but we know this explosion took place here.
I'll simplify it by saying I know an explosion took place
here at location x at time t.
So, this is our crossing. This event is when we crossed.
Then, there is a firecracker. For the firecracker,
I have to give some events [should have said
"coordinates"]. I say it took place at location
x at time t.

What do you say? You measure the distance from
your origin, you call this some x prime,
the time is still t. In Newtonian mechanics,
the time is just the time. How many seconds have passed is
the same for everybody. The question is,
"What is the relation between x prime and x?"
That's what we want to think about.
So, you guys should think before I write down the answer.
What's the relation of x prime to x?
Well, this event took place at time t,
so I know that your origin is to the right by an amount
u times t.

So, the distance from your origin for this event,
I maintain is x prime is x - ut.

Again, I want all of you to follow everything.
These are all simple notions. Our origins coincided at zero
time for the event that occurs at time t.
Therefore, in a sense, you are rushing toward the
event. You've gone a distance
ut. Therefore, the distance from
your origin to the event will be less than mine by this amount
ut. This is the law of
transformation of coordinates in Newtonian mechanics.
If you have an event--if you want, formally,
you can define it time, t prime for the primed
observer. It goes without saying that
t prime and t are the same.
There's no notion of time for me and time for you.
There's universal time in Newtonian mechanics.
It just runs. We can call some time a zero.
Once we have agreed, if you say you and I met at
t = 0 and an explosion took place 5 seconds after our
meeting, it's going to be 5 seconds
after our meeting for me and it's going to be 5 seconds after
the meeting for you. The time difference between two
events is the same for all people.
This is called the Galilean transformation.

What are the consequences of Galilean transformation?
Well, let's look at the fact that x prime is x -
ut.

Remember, everything is varying with time.
So x prime is a function of time and x is a
function of time, if you are watching a moving
particle. Suppose this firecracker is not
just one event, but it's a moving object.
Let's give the object some speed;
it's moving to the right. Then the velocity,
according to me--I'm going to call v as dx/dt is
the velocity. Let's just call it a bullet,
according to S. Then w -- it's the
standard name -- is the velocity of the bullet according to
S prime. So, what I've done is,
I first took one event and I gave it some coordinates and I
told you how to transform the coordinate from one person to
the other person. But now, take that point
x not to be a fixed location but a moving object,
so that as a function of time that body is moving.
Then its velocity at any time is dx prime dt
according to you; that is the dx/dt
according to me, minus the derivative of this,
which is u. Now, does that make sense?
This should agree with common sense.
For example, if that bullet is going at 600
miles per hour to the right, that is 600 for me,
and you are going to the right in your train at 200 miles per
hour, you should measure the bullet speed to be reduced by
200 and you should get 400. That's all it means.
The two people will disagree on the velocity of the bullet
because they are moving relative to each other.
This is the way you will add velocities.
But let's look at the acceleration.
dw/dt is going to be dv/dt - 0 because
u is a constant.

That means you and I agree on the acceleration of the body.
We disagree on where it is. We disagree on how fast the
bullet's moving. But we agree on the
acceleration of the body because all I've done is add a constant
velocity to everything you see. Therefore, if according to you
the velocity of the body is not changing, according to me the
velocity of the body is not changing,
because the constant added will drop out of the difference.
Or, if the body has an acceleration,
we'll both get the same answer for the acceleration.
So, that is the common acceleration a.
So, if you like, a-prime is the same as
a.

So, the acceleration of bodies doesn't change when you go from
one frame of reference to another one going at a constant
speed.

All right, so let's look at F = ma,
which is md^(2)x/dt^(2) is equal to some force on the
body. And you look at the body and
you say d^(2)x prime over dt^(2) is the force on
your body. First, I want to convince you
we wanted to see that the left-hand sides are equal
because the acceleration's the same.
Then, I want to convince you that the right-hand sides are
also going to be equal. I can take many examples but
eventually you will get the point.
Let us not consider one body, but let's consider two bodies.
Two bodies are feeling a certain force due to,
say, gravitation. And gravitation is,
of course, a force in three dimensions but let's write the
force in just one dimension. And let's say the force of
gravity is equal to 1 over x_1 -
x_2. Force on 1 due to 2 and the
force on 2 due to 1 will be minus 1 over x_1 -
x_2. The real forces are separation
in three dimensions but this is a fictitious force.
I want to call it gravity. It is any force that depends on
the coordinates of the two particles.
So, I will say m_1d^(2)x
_1 over dt^(2) is 1 over
x_1 - x_2.
And m_2d^(2)x _2 over
dt^(2) is minus 1 over x_1 minus--I
have forgotten constants like g and
m_1 and m_2.
They don't matter for this purpose.
So, here are two bodies. They feel a force for each
other and I've discovered what the force is.
It's 1 over x_1 - x_2.
I don't care if it's 1 over x_1 -
x_2 or (x_1 -
x_2)^(2); that's not important.
What's important is, it depends on x_1
- x_2. You come along and you study
the same two masses. What will you say is happening?
You will say, m_1d^(2)x
_1 prime over dt^(2) is equal to 1 over
x_1 prime minus x_2 prime.
Maybe I will--I'm sorry. Let me do it a little better.
I can tell you what you will see.
Given this is what I see, I can tell you what you will
see. Let's do that in our head.
We know that the acceleration is the same for any mass so I'm
going to write this thing as m of dx over prime
dt^(2). In other words,
the acceleration according to me is the same as the
acceleration according to you. Then, I'm also going to write
the right-hand side as x_1 prime minus
x_2 prime. Do you understand that?
If there are two bodies feeling a force, if you see it from a
moving train, the distance between the two
bodies is the same for you and me,
because x_1 prime is x_1 -
ut and x_2 prime is x_2 -
ut. Take the difference;
the difference between the location of the particles is the
same for you and me. Acceleration is the same,
mass is postulated to be the same, so I know that you will
get the same law that I get. You will get F = ma;
your acceleration will be the same as mine;
the force you attribute between the two bodies will also be the
same. That is why I know that you
will also deduce the same Newtonian laws that I will.
You can also say it differently. If I woke up from my nap and I
am now in a moving train and I examine the world around me,
I'm going to get the same F = ma.
Because as seen by a person on the ground, the masses obey F
= ma. I am in this moving train now
but I have the same acceleration for each mass and I have the
same force. So, if you want,
I'll complete the second equation.
m_2d^(2) x_2 prime over
dt^(2) will be minus 1 over x_1 prime
minus x_2 prime.
If this is a little difficult, we should talk about this.
I'm telling you that if I deduce F = ma and the
F depends on the separation between the
particles, then I'm sure that you will
find the same laws of motion because the acceleration is the
same that I get because we have seen a-prime is the same
as a. And the force will also be the
same because the force depends on the separation between
particles. And that doesn't depend on
which train you're in or it's not affected by adding a
constant velocity to the frame of reference.
So, if you like, this is the way you prove in
Newtonian mechanics the principle of relativity.
So, not only is it something you observe by going on trains
and what not, you can actually show that this
is the reason everything looks the same.
In other words, if the train was at rest on the
platform and you and I were comparing notes and we both find
F = ma, I go to sleep and I'm waking up
and the train is going at a constant speed,
if you can look through the window and look at the objects
in my train, you will say they obey F = ma because
nothing has happened to you. But you will predict I will
also say F = ma because if you see an acceleration,
I will see the same acceleration.
If you see a distance between two masses to be one meter,
I'll also think it's one meter. If the force is 1 over the
square of the distance, we'll agree on the force,
we'll agree on the acceleration,
we'll agree on everything. And once you've proven F =
ma is valid, it follows that every
mechanical phenomenon will behave the same way.
That's the reason things behave the same way.
Yes? Student: If for some
weird reason, suppose different frames of
reference, the rule F = ma was to fail,
what would happen? Professor Ramamurti
Shankar: You mean if the rule failed in the other frame?
Student: Hypothetically. Professor Ramamurti
Shankar: Yes. Suppose, hypothetically,
that happened. Then, it would mean that when
you wake up in the train, you will look at the world
around you, it will look different because F ≠
ma. You will conclude,
when I went to sleep it was F = ma;
when I got up, F ≠ ma,
the train is moving. So, you will have to conclude
that uniform velocity makes detectable changes.
And if you look outside the train, to the other train,
the other train's going backwards.
You can now no longer say, "You're going the other way,
I'm not moving", because the other person will
say, "Hey, F = ma works for me."
It doesn't work for you. So, you're the guy who's
moving." So, you've lost the equal
status with other inertial observers because those for whom
F = ma worked will say they're not moving and for you,
it doesn't work, so you will have to concede you
are moving. So, uniform velocity,
if it makes perceptible changes, can no longer be
considered as relative. It's absolute and if you and I
find each other moving, there may be a real sense in
which I am at rest and you are moving,
because for me F = ma works and for you it doesn't.
Well, that's not what happens. In real life,
you find it works for both of these and either of us can
maintain we are not moving. So now, you've got to fast
forward to about 300 years. This goes on,
no problem with this principle of relativity and 300 years
later, people have discovered
electricity and magnetism and electromagnetism and
electromagnetic waves, which they identify as light.
And then, it was discovered that what you and I call light
is just electric and magnetic fields traveling in space.
You don't have to know what electro-magnetic fields are
right now. They are some measurable
phenomenon. They are like waves.
And the waves have a certain velocity that Maxwell calculated
and that velocity is this famous number 3 times 10 to the eight
meters per second. And the question was,
"For whom is this the velocity?"
For example, you can do a calculation of
waves on a string, something we'll be able to do
in our course. Waves on a string will be some
answer that depends on the tension on the string and the
mass density of the string and that's the velocity as seen by a
person for whom the string is at rest.
But if you calculate the waves of sound in this room -- I talk
to you, you hear me slightly later -- the time it takes to
travel is the velocity of sound in this room.
That is calculated with respect to the air in this room because
the waves travel in the air. In fact, the fact that all of
us are sitting on the planet, which itself is moving at
whatever, 1,100 miles per second,
doesn't matter, because the air is being
carried along so even if the Earth came to a sudden halt,
as far as the velocity of sound in this room is concerned,
it won't matter, because we are carrying the
medium with us. So, people wanted to know what
is the medium which carries the waves of light,
electromagnetic waves? First of all,
the medium is everywhere because--How do we know it's
everywhere? Can anybody tell me?

Yes? Student: It travels
through the vacuum of space. Professor Ramamurti
Shankar: Right. It travels in the vacuum of
space. We can see the Sun;
we can see the stars, so we know the medium is
everywhere. Then, you can sort of ask,
"How dense is the medium?" It turns out that the denser
the medium, the more rapidly signals travel,
in most of the things that we know.
For example, when we look at waves in sound
and when we look at sound waves in a solid,
or in iron, you find in a very dense material,
that the velocity is very high. S,o this medium,
which is called "ether," would have to be very,
very dense to support waves of this incredible velocity.
But then, planets have been moving through this medium for
years and years and not slowing down.
It's a very peculiar medium. But it has to be everywhere so
we are all immersed in this medium because we are able to
send light signals to different parts of the universe.
And the question is, "How fast is the Earth moving
relative to this medium?" You understand?
This medium is all pervasive. We know that we can see the
stars so it's going all the way up to the stars and beyond.
And we are immersed in this and we are drifting around in space.
What is our speed relative to the medium?
That's the question that was asked.
Well, to find the speed relative to the medium,
you calculate the velocity in the medium by Maxwell's theory.
So, here's the medium in which waves travel at a certain speed.
This is planet Earth going around the Sun.
At some instant, you will have a certain
velocity with respect to ether. And therefore,
the velocity of light as seen by you will be modified from
c to c - V. In particular,
suppose the waves are traveling to the right in ether.
Let me draw it this way. The Earth is going at this
instant at the speed V. We expect the speed to be c
- V because part of the speed is neutralized because you
are going along with the waves. You'll see a slower velocity.

So, Mr. Michelson and his assistant
Morley--they did the experiment. And they got the answer equal
to c. What does that mean?
Student: The speed of light [inaudible]
Professor Ramamurti Shankar: No,
no, but you cannot jump to that right now.
If you are following Newtonian physics, your expectation is,
it should be c - V. Yes?
Student: It means that there is no ether.
Professor Ramamurti Shankar: Well,
that's--not so fast, but it certainly means the
following. Well, there's a simpler answer
than that. Yes?
Student: It means that the Earth is moving with respect
to the ether. Professor Ramamurti
Shankar: At what speed? Student: Zero.
Professor Ramamurti Shankar: Zero.
Because you don't have to--Look, you guys are ready to
overthrow everything because you know the answer.
But you've got to put yourself in the place of somebody in the
early 1900s. There's no reason to overthrow
anything. The answer is,
you're going at the speed zero. Of course, you realize,
that it is incredibly fortunate that on the one day Michelson
wants to do the experiment, we happen to be at rest with
respect to the ether. Fine.
But we know that luck is not going to last forever because
you are going around the Sun. On a particular day may be.
But that velocity was such that on that day the Earth was at
rest with respect to the ether. It's clear that six months
later, when we are going the other direction,
you cannot also be at rest with respect to the ether.
But that's what you find when you do the experiment.
You find every day you get the same answer and you jolly well
know you are not at rest. You are moving around the Sun
for sure. Yes?
Student: Did they postulate a drag for the Earth
turning [inaudible] Professor Ramamurti
Shankar: Yes. So, people tried other
solutions. But it is simply a fact that
when you move one way or six months later in the opposite
way, you get the same answer c.
So, one possibility is, you don't want--Look,
don't be ready to do revolutions, try to avoid it.
So, one answer is, look at the speed of sound.
You and I talked to each other then and we talked to each other
six months from now; we get the same speed of sound.
The speed of sound is published in textbooks,
right? Seven hundred and something
miles per hour. How come that doesn't change
from day to day? Anybody here on this side can
tell me why the speed of sound doesn't change from day to day
even though we are moving? No one here can guess?
Student: [inaudible] Professor Ramamurti
Shankar: No, we are moving.
Even six months from now, we get the same speed of sound
in this room. When I talk to you,
does it matter what time of year it is?
Student: The medium is moving along with us?
Professor Ramamurti Shankar: Yes,
we are carrying air. As the Earth moves through
space, it carries the air with you and the speed of the wave is
with respect to the medium. If you can carry the medium
with you, then it doesn't matter how fast you're moving.
So, they tried that. They tried to argue that the
Earth carries ether with it the way it carries air with it.
Then, it's not an accident you are at rest with respect to the
ether because you're taking it with you.
But it's very easy to show by looking at distant stars that
you cannot be doing that. I don't have time to tell you
why that is true. So, you cannot take the ether
with you and you cannot leave it behind, and that's the impasse
people were in. So, it's as if there's a car
that's going to the right at a certain speed c.
You move to the right at some speed, maybe c/2.
I expect you to get a speed c/2.
But you keep getting c. You go three fourths of the
velocity of light; you still get the velocity of
light. That is very contrary to what
we believe. In fact, that's in violent
opposition to this law here. If this V were not a
bullet but a light beam, suppose for me traveling at a
speed c and you're traveling to the right at speed
u, you should get c - u.
That's the inevitable consequence of Newtonian
physics. And you don't get that.
And that was a big problem. So, people tried to fix it up
by doing different models of ether, none of which worked.
And nobody knew why light is behaving in this peculiar
fashion, so that's when Einstein came in and said,
"I know why light is behaving in this peculiar fashion.
It is behaving in this way because if it didn't behave in
this way, if the speed of light depended on how fast I'm moving,
then when I wake up in this train, all I have to do is
measure the speed of light. Originally, I got some number;
now I get to get a different number and the difference will
give me the speed of the train. So, it would have been possible
to detect the velocity of the train without looking outside,
just by doing an experiment with light.
So, even though mechanical laws involving F = ma are the
same, laws of electricity and magnetism would be such that
somehow they would betray your velocity.
And that would mean uniform velocity does make an observable
change because it changes the velocity of light that you would
measure. " But conversely,
the fact that you keep getting the same answer means that
electric and magnetic phenomena are part of the conspiracy to
hide your velocity. Just like mechanical phenomena
won't tell you how fast you're moving, neither will
electromagnetic phenomena. Because to Einstein,
it was very obvious that nature would not design a system in
which mechanical laws are the same but laws of electricity are
different. So, he postulated that all
phenomena, whatever be their nature, will be unaffected by
going to a frame at constant velocity relative to the initial
one. That's a very brave postulate
because it even applies to biological phenomena about which
I'm sure Einstein knew very little.
But he believed that natural phenomena will just follow
either the principle of relativity or they won't.
And that is something you should think about.
Because that was the only reason he had.
He just said, "I don't believe chapters 1
through 10 in our book obey relativity and chapters 20
through 30 where we do E&M doesn't."
These are all natural phenomena that will obey the same
principle, which says all observers that are uniform
relative to motion are equivalent.
Now, that's really based on a lot of faith and even though
scientists generally are opposed to intelligent design,
we all have some bias about the way natural laws were designed;
there's no question about it. You can talk to any practicing
physicist. We have a faith that underlying
laws of nature will have a certain elegance and a certain
beauty and a certain uniformity across all of natural phenomena.
That is a faith that we have. It's not a religious issue;
otherwise, I wouldn't bring it up in the classroom,
but it is certainly the credo of all scientists,
at least all physicists, that there is some elegance in
the laws of nature and we put a lot of money on that faith,
that the laws of nature will do this and will not do this.
Who are we to say that? Who are we to say nature
wouldn't have a system in which mechanics obeys the laws but
electricity and magnetism doesn't?
We haven't run into somebody called nature.
We don't worship a certain deity called nature but we
believe the laws of nature obey that.
So, even though scientists or physicists in particular may not
believe in design by any personal God,
they do believe in this underlying, rational system that
we are trying to uncover. You could be disproven,
you could be wrong in making the assumption,
but here it was right. It was really driven by this
notion that all laws of physics should obey the same principle
of relativity. So, Einstein's postulates are
that light behaves in this way because if it didn't behave in
this way, it would violate the principle
of relativity, whereas we know mechanical
phenomena do and electrical phenomena would not and that
cannot be the case. You have a question?
Student: Why would that not apply to the speed of sound?
Professor Ramamurti Shankar: Yes.
Because in the case of the speed of sound,
you can take the medium with you;
there is no such experiment you could perform.
See, in the train, if you could carry the ether
with you, there's no surprise you would get the same answer.
But we know we cannot carry the medium with us,
that comes from extra-terrestrial experiments.
That's why the velocity of sound is not elevated to a
fundamental velocity on which everybody will agree.
So, the two great postulates--You've got to know
where the postulates come from. Postulate Number 1 is simply a
restatement of the relativity principle.
I'll just say it in one sentence.
Exact wording is not important. All inertial observers are
equivalent.

"Equivalent" means each one of them is as privileged as any
other one to discover the laws of nature.
The laws of nature, we found, are not an accident
related to our state of motion. If I find some laws,
and you're moving relative to me, you'll find the same laws.
And if you and I find each other in relative motion,
you have as much right to claim you are at rest and I am moving
and I have as much right to claim that I am at rest and you
are moving. There is complete symmetry
between observers in uniform relative motion.
There is no symmetry between people in non-uniform [relative]
motion. As I said, non-uniform motion
creates effects which can destroy me and not destroy you.
So, no one's trying to talk their way out of acceleration,
whereas you can talk your way out of uniform velocity.
That's the first principle. So this was there even from the
time of Newton. What is true here is that all
inertial observers are equivalent with respect to all
natural phenomena, meaning all natural laws.

And that is a generalization, when we say "all" instead of
just mechanical.

And the Second Postulate, you call it a postulate because
there is just no way to deduce this,
is that the velocity of light is independent of the state of
motion of the source, of the observer, of everything.

If a light beam is emitted by a moving rocket,
it doesn't matter. If a light beam is seen in a
moving rocket, it doesn't matter.
All people will get the same answer for the velocity of
light. Student: Is there a
reason why the speed of light is constant?
Professor Ramamurti Shankar: No.
That's why it is a postulate. You can show a few things later
on. You can show that if there is
any other speed, which is the same for
everybody, that would have to be the speed of light.
In the final theory of relativity, there are not two or
three velocities that come out the same for everybody.
There is only one velocity that can have the same answer for all
people. That velocity is the velocity
of light. By that I don't mean it has to
be light itself. For example,
gravitational waves travel at the speed of light.
It's not just the light. It has to do with the velocity
of light being a single number which has to have the same value
for everybody. Okay, so it looks like he has
solved a big problem because he has said why light behaves this
way; light behaves this way because
it is part of the big conspiracy to hide uniform motion.
But you will see that you have made a terrible bargain because
once you take these two postulates,
you have restored the relativity principle to all
phenomena. Okay.
You've gone beyond mechanical phenomena to electro-magnetic
phenomena. But you will find that you have
to give up all the other cherished notions of Newtonian
physics. Think about why.
We are saying, here is a car going at 200
miles an hour, according to me.
You get into your own car and follow that car at 50 miles an
hour. You should get 150 but you keep
getting 200. This may not be true for cars
going at the speeds I mentioned but when finally you are talking
about a pulse of light, that is true.
And you've got to agree that is really not compatible with our
daily notions or with the formula I wrote down,
w = v - u. When you put v = c,
w has got to come out to be c.
And that's not a property of the Newtonian transformation.
So, what we are looking for is a new rule for connecting
x and t and x prime and t
prime, such that when the velocities
are computed and applied to the velocity of light,
you get the same answer. That's what we want to do now.
So, here is how we are going to do this.
Now, let's think about it. Let me send a pulse to the
right at speed c. You are going to the right at
three-fourths of c. My Newtonian expectation is,
you should get the speed of the pulse to be one-fourth of
c. But you insist it is c.
So, what will I say to you? What will I accuse you of doing?
I say you should get c/4 and you're getting c.
And you're finding velocity by finding the distance it travels
and dividing by the time so you are jacking up a number like
one-fourth c to c itself.
So, what could make you do that? Yep?
Student: If I perceive that you're moving forward,
or you're contracting your length,
then you're going to measure velocity keeping in mind you
should have had a greater length to begin with.
Professor Ramamurti Shankar: Yes.
So, one option--Let me repeat what he said.
I will say your meter sticks are being somehow shorter.
When you and I were buddies and were on the same train,
we agreed on the meter stick. But now we have gone on the
moving train. I will say there is something
wrong with your meter stick. Not only something wrong.
Specifically, I would say your meter sticks
have shrunk. For example,
if they have shrunk to one-fourth their size,
it is very clear that you would get a velocity of four times
what I expect. But there's another possibility.
Student: Time may be running slow.
Professor Ramamurti Shankar: Your clocks may be
running slow. So, you let the light travel
for four seconds and you thought it was only one second.
That's why you got four times the answer;
or it could be both. But something has to give.
And that is why it is an amazing theory.
That's why it is also amazing to me that somebody who was 26
years old would simply follow the consequence of this theory
and take it wherever it takes you but it is at the very
foundations of space and time that you have to modify.
So, even though you restored the relativity principle and
brought it back to the front, the price you have to pay is to
give up your notions that length is length and time is time.
We used to think a meter stick is a meter stick and a clock is
a clock. If I have a clock that ticks
out one second, you take it on a train,
I expect it to be ticking one second, but we're saying it's
not. So, something has to give in
length measurement or time measurement or both.
And that's what we're going to find out.
So, here's how you find that out.
Let us say that--Maybe I'll do it all on one blackboard because
this is the key to the whole calculation.
You remember now, if there is an event here,
you call it x prime and I call it x.

And according to me, you have traveled the distance
ut.

So, x prime equals x - ut is what we used to say
in the old days. And the converse of that is
x = x prime + ut. But now, we'll admit the fact
that maybe t and t prime are no longer the same.

But that's not all we will do. I will say, whenever you give
me a length x prime, I just don't buy it.
I take any length you give me and I jack it up by a factor of
γ [correction: should have said 1/γ]
to get the length according to me.
And you will take any answer I give you and you will multiply
it by the same factor. Don't worry about this yet!]
In other words, we don't buy our units of
length, so if you say it should be
x prime + ut prime, that's your formula
backwards for me. The coordinate of the event,
according to you, in the old days,
was x prime + ut. Now, we admit that t and
t prime may not be the same.
Then we say, but I will not take your
lengths, I will multiply them by γ to get the lengths
according to me. And you will not take my
expectations, but you will multiply it by the
same γ; that is the symmetry between
the two observers. In other words,
if I think your meter sticks are short and γ
[correction: should have said 1/γ]
is a number less than 1, I'm allowing you to accuse me
of having meter sticks which are short.
It is very interesting. If I said your meter sticks are
short and you say my meter sticks are longer than average,
that's an absolute difference. But we both accuse each other
of using shortened meter sticks, and so we use the same factor
of γ. We are going to find this
γ now. Student: How do you know
that both the distance and time are different?
Professor Ramamurti Shankar: We'll give it the
possibility that they are different and then we will see
that they are different. We know something's wrong with
space and something's wrong with time.
So, we'll not assume that t prime is equal to
t. You have to open up the
possibility. In the end, it may be that the
nature will say t prime is t and something
happens to length alone. But we'll find the answer is
more symmetric. Student: Why do we say
that the symmetry is the same? Professor Ramamurti
Shankar: Because that is the symmetry between the two
observers. If I want to say your meter
sticks are short, why should you concede that?
You should be able to accuse me of saying--The only difference
between you and me is you are moving to the right and I'm
moving to the left. Other than the sign of the
velocity, each person says the other person is moving and so we
will say that any length you give I'll discount by a factor
of γ. By symmetry,
anything I call a length, you will discount by the same
factor of γ. Now, let's apply this.
(x,t) was a certain event,
right? Let's apply it to the following
event. You have to follow this very
carefully. When you and I crossed,
remember that was at the origin of coordinates,
x equals to t equals to zero and x
prime equals to t prime equals to zero.
At that instant when our origins touched,
let us emit a flash of light. Okay, maybe when the origins
touched, there's a spark, the light signal goes out and
the light signal is detected here, by a light detector.
That second event, detection of the pulse,
it has a coordinates (x,t) for me;
it has a coordinates (x prime, t prime) for you.
The same event is given different coordinates.
We've already used the fact that coordinates will be
different but we are saying not only will x prime not be
equal to x; t prime may not be equal
to t either.

Okay, but now let's write down one important condition.
What is the relation between x prime and t
prime in this particular pair of events?

What is x prime? It is the location of the light
pulse after certain time? So, what's the relation of the
location of the detection of the light pulse and the time
t prime? Yes?
Student: [inaudible] Professor Ramamurti
Shankar: He is saying x prime--Do you guys
understand that? Do you agree with that
statement? This is not a random event.
The second event was the detection of a light pulse.
Light pulse left the origin t prime seconds earlier
and has come to this point, according to this guy,
and the ratio of the distance to the time is the velocity of
light. But it is also true that for
me, the light went to distance x in a time t.
So, that's the relation of x to t for me
also. I'm going to use those two
results and combine it with this to find this factor γ
and we will do that now. I want you to multiply the
left-hand side by the left-hand side and the right-hand side by
the right-hand side of that equation.
I hope you understand that in the Galilean days,
in the old days--Let's see what you will say.
I will say, x prime is x - ut because the
origins have shifted by an amount ut.
And you will say x is x prime plus ut
with the same time. Now, I'm saying time is
different. Not only that,
I don't buy your length. If you expect me to have this
length, I say "no." You exaggerated everything.
I'll scale it down by γ and vice versa.
So, if you multiply the left-hand side by the left-hand
side, you get xx prime, the right-hand side you get
γ^(2) times xx prime plus uxt prime
minus ux prime t minus u^(2)tt prime.

Now, divide everything by xx prime.
Then, you get 1 = γ^(2) times 1 + u.
If you divide this by xx prime, you'll get t prime
over x prime. If you divide this by xx
prime, you'll get t/x. And here you get u^(2)
times t/x, times t prime over
x prime.

So, what does that mean? Well, t prime over
x prime is 1/c and t/x is also
1/c because of what I wrote here.
If you want, let me write this as t
prime over x prime equals the c and t over
x equals the c. So, they will cancel.
And I'll get 1 = γ^(2) times 1 - u^(2) over
c^(2) because this t/x is a 1/c and
t prime over x prime is another 1/c.

So, that gives us the result that γ is 1 over square
root of 1 - u^(2) over c^(2).

If you plug the γ back in, you will find x prime
is x - ut divided by [square root of]
1 - u^(2) over c^(2).

Now, once you do that, once you got the relation
between x prime and x, you can go to the
lower equation and solve for t prime.
I don't feel like doing that. It's a simple algebraic
equation once you've got x prime how to solve for
t prime. Take the second of the two
equations and solve for t prime.
That detail I won't fill out, but you will get t prime
is t - ux over c^(2) divided by this.
So, I've not done every step but I've given you all the
things you need to do the one step.
There are equations up there that relate x and
x prime to t and t prime so if you can get
x prime in terms of x and t,
the other equation that you solve will give you t
prime in terms of x and t.
And this is the result.

Okay. This guy deserves two boxes
because it is the greatest result from relativity;
it's called the Lorentz transformation.
And we've been able to derive the Lorentz transformation with
what little we know. And you can see,
you can be a kid in high school and you can do this.
There's no calculus or anything else involved,
other than being open to the fact that the velocity of light
behaves in this strange way. Yes?
Student: How do you get t/x = c?
Professor Ramamurti Shankar: Why is t/x =
c? Oh, of course,
you're quite right. So, you caught the mistake here.
t/x is 1 over c and 1/c.
I really meant to write x = ct.
Yes? Student: [inaudible]
Professor Ramamurti Shankar: No.
If you define γ to be the absolute value by which you
transfer lengths from you to me, then you can take the positive
root. Student: [inaudible]
Professor Ramamurti Shankar: Well,
I can also tell you other reasons.
Let's take this formula here. Let's take the case where the
velocity is very small compared to the velocity of light.
That u/c is a very small number.
This number is almost 1 and I get x prime because x
- ut, which I know to be the correct answer at lower
velocities. If you pick the minus sign,
I'll get minus of x - ut, that's not the right
answer, not even close to the right answer.
At low velocities, if you go to velocity u
over c much less than 1, you have got to get back to
Galilean transformation. You can see if u over
c goes to zero. You can forget all about this
factor here. You get x prime is
x - ut and here, u over c can be
neglected, forget all that,
you get t prime equals t.
So, this coordinate transformation would reduce to
the Galilean transformation if the velocity between me and you
is much smaller than the velocity of light.
So, the formula really kicks in for velocities comparable to
velocity of light. Yes?
Student: [inaudible] What happens if u >
c? Professor Ramamurti
Shankar: Well, you start getting crazy
answers, right? You can already see that the
theory will not admit velocities bigger than the velocity of
light. You can already see it in this
formula. That tells you that the one
single velocity that you wanted to be the same for everybody is
also the greatest possible velocity;
that no observer can move at this speed with respect to
another observer that is equal to what is in excess of the
speed of light. So, the speed of light,
which came out to be a constant in the beginning of the theory,
is also turning out to be the upper limit on possible
velocities. That's the origin of the
statement that no observer can travel at a speed bigger than
light, but we'll discuss it more and more.
But you have to understand what it is that is being derived,
what is the meaning of this formula here.
What is it telling you? If I say, this is called the
Lorenz transformations, what do they tell you?
What are these numbers and what's their significance?
Would you like to try? Student: Well,
one thing that they tell you is if u happens to be
greater than c [inaudible]
Professor Ramamurti Shankar: No,
no, I don't mean what happens in the formula in special cases.
What is it relating? What is xt and what is
x prime t prime? Student:
Okay, so x prime would be the distance that the person
who's traveling at the higher speed experiences.
And [inaudible] and so for that person,
distance is going to seem shorter?
Professor Ramamurti Shankar: No.
See, I'm not even telling you to get consequences of the
equation. What are the numbers x
and t in this equation? And what are the numbers
x prime and t prime in this equation?
Student: x and t are distance and time
for a person who is in an inertial frame of reference,
who is not moving. Professor Ramamurti
Shankar: Right. Student: And x
prime and t prime are distance and time for the person
who is moving at the speed of [inaudible]
Professor Ramamurti Shankar: And when you say
distance and time, what do you mean,
distance and time? Student: The way that
the length of the distance will seem to them that they travel.
Professor Ramamurti Shankar: But what is
happening at xt; it's the coordinates of what?
Student: Of their location [inaudible]
Professor Ramamurti Shankar: They are located at
zero, zero. Right?
What's happening at x and t?
Yes? Student: It's observing
the event. Professor Ramamurti
Shankar: It's the event. The key I was looking for is an
event. You've got to understand what
the formula is connecting. Things are happening in space
and in time, right? Something happens here.
That something has a spatial coordinate and a time
coordinate, according to two observers.
The observers originally had their origins and their clocks
coincide when they passed; that's how they're related.
And one is moving to the right at speed u.
Then, the claim is that if one event had a coordinate x
and t for one person, for the other person moving to
the right at speed u, the same event would have
coordinates x prime and t prime and the relation
between x and t and x prime and t
prime is this. Yes?
Student: The two observers [inaudible]
don't they observe different laws and [inaudible]
Professor Ramamurti Shankar: No.
The fact that an event has different coordinates doesn't
mean that you are observing different laws.
For example, let's take that fire
extinguisher. We look at it,
it's obeying F = ma, right?
The coordinates of the fire extinguisher with me as the
origin is quite different from you as the origin.
Student: [inaudible] Professor Ramamurti
Shankar: You mean in these new equations?
Yes, in these new equations, F = ma will not work;
that's correct. Student: They are
inertial references, even though they are moving?
Professor Ramamurti Shankar: Ah.
Yes. So, the point is the laws of
Newton themselves have to be modified.
F = ma will be modified in a certain way but the new
modified laws will reduce to F = ma at low velocities,
which is why in the old days it looked like F = ma.
But there will be new laws, but they will also have the
property that when I measure them I'll get the laws that will
agree with what you measure. Yes?
Student: Does individual values for time or distance
still have to agree? Professor Ramamurti
Shankar: The coordinates of an event will differ from person
to person. That's not the same as saying
the laws as deduced will be different.
For example, there are two stars which are
attracted to each other by gravitation and they are
orbiting around their common center of mass.
If I see them, I will find that they obey the
law of gravitation with m_1m
_2 over r^(2) where r is
the distance between the points and the acceleration is whatever
I think it is. You can go on a rocket and look
at the same two stars. They will be in a more
complicated motion, maybe the whole system will be
drifting a little to you, but their acceleration will be
the same as what I get and the force between them will also be
the same as what I get and the laws that you would deduce by
looking at that star would be the same laws that I would
deduce. So, that's a difference between
the laws being the same and the coordinates being the same.
No one said x prime and x are the same in that
equation there. They are different.
We are looking at it from different vantage points.
But the fact is that force is equal to mass times acceleration
is the same for the two people. Okay.
The laws will be the same but things won't look the same.
For example, you can stand on your head,
don't even have to go to another frame of reference;
you can stand on your head, your z coordinate is the
minus of my z coordinate; to every point I give a
z, you will give a minus z.
But the world, even though you are a little
messed up and want to stand on your head,
you have every right to do that and you will find that F =
ma. So the point is,
the way we see events may depend on their origin or
coordinates, but the laws we deduce are to
be distinguished from the perception that we have.
Okay? For example,
if I'm on the ground, I send a piece of chalk,
it goes up and it goes down. If you see me from a moving
train, you would think it went on a parabola.
So, no one says the chalk will go up and down for you.
For you, it will go like this but its motion will still obey
F = ma, is what I'm saying.
That's all you really mean by saying things look the same.
So, what you have to understand is that Lorentz transformations
are the way to relate a pair of events, given events.
Here's a simple example. If you live in the xy
plane, there's a point here. It's not an event,
it's simply a point in the xy plane.
You measure it this way and that way and you call it the
coordinates. If somebody else picks a
different coordinate system with an angle θ,
that person will say that's x prime and that is
y prime and x prime and y prime are not
the same. I remind you,
x prime is x cos> θ [delete "minus
y"] plus y sin θ,
etc., and y prime is something else.
θ is the angle between the two observers.
So, the point is the point. It certainly looks different to
the two people, but the same point has two
coordinates. Similarly, the same event,
like the collision of two cars, will have different events for
different people. That's not the new part.
The new part is that the rules for connecting xt to
x't', is quite different from the Galilean rules,
new rules. It's what you guys have to
understand. And finally,
why did Einstein get the credit for turning the world into four
dimensions instead of three? After all, x and
t were present there, too.
The point is, t prime is always equal
to t no matter how you move,
whereas in the Einstein theory, x and t get
scrambled into x prime and t prime just the way
x and y get scrambled into each other when
you rotate your axis. So, to have time as another
variable that doesn't transform at all is not the same as making
it into a coordinate. The four-dimensional world of
Einstein is four-dimensional because space and time mix with
each other when you change your frame of reference.
That's what makes t now a coordinate as previously it
was something the same for all people.